Projection algorithm [1]} is a kind of classic while effective iterative solver for computing an approximate solution for (REF ). As we know, many existing practical projection iterative algorithms are under the framework of Petrov-Galerkin conditions, such as the Kaczmarz and coordinate descent (CD) methods, see [2]}, [1]}, [4]}, [5]}, [6]} and the references therein. Let the constrained subspace and the search subspace correspond to \(\mathcal {L}={\rm span}\lbrace  Y\rbrace \)  and \(\mathcal {K}= {\rm span}\lbrace  Z\rbrace \) , respectively, where \(Y\)  and \(Z\)  are two parameter matrices. It leads to the following iteration scheme [2]}, [6]}.
\(^{(k+1)} = ^{(k)} + Z(Y^T A Z)^\dag Y^T (-A^{(k)})\) 
